This book aims to make the subject of geometry and its applications easy and comfortable to understand by students majoring in mathematics or the liberal arts, architecture and design. It can be used to teach students at different levels of computational ability and there is also sufficient novel material to interest students at a higher cognitive level. While the book goes deeply into the applications of geometry, it contains much introductory material which up to now may not have been known to the student. The constructive approach using compass and straightedge engages students, not just on an intellectual level, but also at a tactile level. This may be the only rigorous book offering geometry that attempts to engage students outside of the mathematics discipline.

Preface xiii

Chapter 1. Triangle and Square Grids 1

1.1. Introduction 1

1.2. Triangle-Circle and Triangle Grids 1

1.3. Square-Circle and Square Grids 5

1.4. Star Designs Based on the Triangle Grid 6

1.5. Star Exploration 10

1.6. The Two Great Systems of Ancient Geometry 10

Chapter 2. The Pythagorean Theorem 13

2.1. Introduction 13

2.2. Five Proofs of the Pythagorean Theorem 13

2.3. Pythagorean Triples and the Brunes Star 18

2.4. Application of the Pythagorean Theorem 19

Chapter 3. Scan Converting of a Line Segment 21

3.1. Introduction 21

3.2. Lines 22

3.3. Lines and Calculus 23

3.4. Scan Converting of a Line 24

3.5. An Example of Scan Converting 24

Chapter 4. Compass and Straightedge Constructions 27

Part 1: The World Within a Triangle 27

4.1. Geometric Constructions 27

4.2. The Fundamental Constructions 28

4.3. The World Within a Triangle 31

4.4. Special Points in an Equilateral Triangle 31

4.5. A Theorem about Inscribed Circles Within a Triangle 32

4.6. Inscribing Regular Polygons in Circle 32

4.7. The Centroid of a Polygon 33

4.8. Additional Constructions 34

Chapter 5. Congruent Triangles 35

5.1. Introduction 35

5.2. The Sum of the Angles of a Triangle 36

5.3. Conditions for Congruent Triangles 36

5.4. Triangle Inequality 37

5.5. Constructing a Triangle from Partial Information 37

5.6. Two Applications of Congruent Triangles 39

Chapter 6. Introduction to Trigonometry 41

6.1. Introduction 41

6.2. The Trigonometry of a Right Triangle 41

6.3. Given the Value of One Trigonometric Function to Derive the Others 43

6.4. Trigonometry of a Triangle with Angles Greater than 90 Degrees 44

6.5. Periodic Functions and their Inverses 46

6.6. Application of Trigonometry to Compute the Angles and Sides of a Triangle Given Partial Information About the Triangle 47

6.7. Calculation of SSA 51

6.8. Additional Problems to Solve Triangles with Partial Information 52

6.9. An Application of Trigonometry to Finding the Radius of Earth 53

Appendix 6A. Proof of the Law of Sines 54

Chapter 7. The Art of Proof 57

7.1. Introduction 57

7.2. Three Important Theorems 58

7.3. Converses 58

7.4. Equivalence Tautologies and Definitions 59

7.5. Proof 59

Chapter 8. Angle 65

8.1. Introduction 65

8.2. Interior Angles of a Triangle 66

8.3. The Exterior Angles of a Triangle 66

8.4. A Useful Relationship Between Interior and Exterior Angles 67

8.5. Interior Angles of a Regular Polygon 68

8.6. Vertical Angles 69

8.7. Radians and Degrees 70

8.8. Angles Within a Circle 70

Appendix 8A. A Theorem of Importance to Non-Euclidean Geometry 71

Chapter 9. Voronoi Domains 73

9.1. Introduction 73

9.2. Construction of Voronoi Domains 74

9.3. Voronoi Domains and Pattern Recognition 78

9.4. Additional Problems 80

Chapter 10. Parallel Lines 83

10.1. Introduction 83

10.2. Euclid's Parallel Axiom 84

10.3. Parallel Lines in Euclidean Geometry 85

10.4. Conditions for Two Lines Being Parallel 86

10.5. The Sum of the Angles of a Triangle 88

10.6. An Application of Parallel Lines to Finding the Circumference of the Earth 88

Appendix 10A. The Geometry of Lines in the Poincare Plane 90

Chapter 11. Bracing a Framework 93

11.1. Introduction 93

11.2. Frameworks 93

11.3. An Algorithm to Determine Rigidity of a Framework 95

11.4. An Introduction to Graph Theory 96

11.5. Determining Rigidity by Graph Theory 98

Appendix 11A. Proof of the Graphical Method for Determining the Rigidity of Frameworks 99

Chapter 12. Similarity 101

12.1. Similarity 101

12.2. Similar Triangles 101

12.3. Surgery on a Right Triangle 103

12.4. The Pythagorean Theorem 104

12.5. Chords, Secant Lines and Tangents to Circles 105

12.6. Problems 107

Chapter 13. Compass and Straightedge Constructions 111

Part 2: Doing Algebra with Geometry 111

13.1. Introduction 111

13.2. Construction of nl, a＋b, and a-b 111

13.3. Construction of a-b and ab 111

13.4. Line Chopper 112

13.5. Doing Algebra with Geometry 113

13.6. Square Roots 114

13.7. Construction of a Sequence of Rectangles with Proportions √n:1 115

14. Area 117

14.1. Introduction 117

14.2. Axioms of Area 117

14.3. Area of a Parallelogram 118

14.4. Area of a Triangle 118

14.5. Area of a Polygon 119

14.6. Area of a Trapezoid 119

14.7. Area of an Equilateral Triangle 120

14.8. Areas of a Family of Similarity Figures 120

14.9. Pythagorean Theorem 122

14.10. Geoboards 123

14.11. Problems 124

15. Vectors and Geoboards 125

15.1. Introduction 125

15.2. What is a Vector? 125

15.3. The Magnitude of a Vector 125

15.4. The Sum and Difference of Vectors 126

15.5. Triangles, Parallelograms and Vectors 128

15.6. Problems 130

15.7. The Law of Cosines 130

15.8. Perpendicular Vectors 131

15.9. Invariance of the Length of a Vector and the Angle between Vectors Under Rotation 132

15.10. Another Way to Name Vectors 133

Appendix 15A. Proof of the Law of Cosines 134

16. Logarithmic Spirals 137

16.1. Introduction 137

16.2. From Right Triangle to Logarithmic Spiral 138

16.3. A Simple Way to Generate a Dense Set of Points on a Logarithmic Spiral 141

16.4. The Law of Repetition of Ratios 142

16.5. Whirling Squares 144

16.6. Additional Constructions 145

Appendix 16A. The Four Turtle Problem 145

Appendix 16B. The Baravelle Spiral 146

17. The Golden and Silver Means 147

17.1. Introduction 147

17.2. The Golden Mean and the Golden Section 147

17.3. The Silver Mean 150

17.4. The Golden Spiral 151

17.5. A Golden Rectangle 152

17.6. A Regular Pentagon 153

17.7. Golden Triangles 153

17.8. Application of the Golden Section to Art and Architecture 154

17.9. A Regular Decagon and Regular Pentagon Within a Circle 155

17.10. Sacred Cut 156

18. Isometries 159

18.1. Introduction 159

18.2. Isometries 159

18.3. Translations 159

18.4. Rotations 160

18.5. Reflections 161

18.6. Glide Reflections 161

18.7. Identity Transformation 162

18.8. Compounding Isometries 162

18.9. Proper and Improper Isometries 163

18.10. Additional Problems 165

Appendix 18A. Proof that an Isometry is Determined by the Transformation of Three Points 166

19. Isometries and Mirrors 169

19.1. Introduction 169

19.2. Reflections in One Mirror 169

19.3. Reflections in Two Mirrors 169

19.4. Reflections in Three Mirrors 170

19.5. Multiple Rotations 173

Appendix 19A. Proof that Any Isometry Can be Carried Out with No More than Three Mirrors 176

Appendix 19B. Three Mirrors in a Configuration of that Results in a Glide Reflection 177

20. Kaleidoscopes and Symmetry 179

20.1. Introduction 179

20.2. Exercises 179

21. Groups and Kaleidoscope Symmetry 183

21.1. Groups and Symmetry 183

21.2. Symmetry of a Square 183

21.3. Multiplications of Symmetry Transformations 185

21.4. Symmetries and Mirrors 187

21.5. Symmetry Groups as an Algebraic System 187

21.6. Group Theory 189

Appendix 21A. Multiplication Table for the Symmetries of a Square 189

22. Frieze Patterns 191

22.1. Introduction 191

22.2. Explanation of the Seven Frieze Patterns 191

22.3. Identifying Frieze Patterns 193

22.4. Additional Problems 194

23. An Introduction to Fractals 197

23.1. Introduction 197

23.2. The Koch Snowflake 198

23.3. A Fractal Tree 199

23.4. A Moonscape 199

23.5. A Cauliflower 200

23.6. A Fractal Wallhanging 200

23.7. Finding the "DNA" of an Iterative Function System 203

23.8. Symmetry Finder 203

23.9. Construction Fractals from the Iterative Function System 206

24. Isometries and Matrices 207

24.1. Introduction 207

24.2. Matrices for Rotations and Reflections in the Plane 207

24.3. Transformations of Mr. Flatlands 210

24.4. Homogeneous Coordinates 212

24.5. Translations 212

24.6. Scaling 214

24.7. Fractals 216

24.8. Projections 218

Appendix 24A. An Introduction to Matrices 221

Appendix A: Projective Geometry Constructions 223

A1. Introduction: Projective Transformations 223

A2. Pappus' Theorem 226

A3. The Dual of Pappus' Theorem 227

A4. Pascal's Theorem 228

A5. Branchion's Theorem 229

A6. Construction of a Conic Section 230

A7. Desargues Theorem 231

A8. A Hexagonal Net 232

A9. Quadrilateral Net 233

A10. Points of the Quadrilateral Net Going to Infinity I 235

A11. Points Going to Infinity II 236

A12. Pole and Polar of an Ellipse or Circle 238

A13. Inversion in a Circle 239

Appendix B: Glossary 243

Appendix C: Essays 249

Bibliography 251

Index 253

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